Slide 1

Slide 2 Methods of Proofs October 20, 2014 1 Slide 3 A Good Proof State your plan Avoid excessive symbols Simplify as much as possible Good notation 2 Slide 4 A Bad Proof We dont prove by: Obviousness: The proof is so clear that it need not be mentioned Imagination: Well, let it be true Convenience: It would be nice if it were true, Intimidation: Dont be stupid; its of course true! Lack of sufficient time: Because of the time constraint, http://jwilson.coe.uga.edu/EMT668/EMAT6680.F99/Challen/proof/proof.html 3 Slide 5 Class Discussion Whats wrong is the following proof of 1=2? 1. a = b 2. a 2 = ab 3. a 2 - b 2 = ab - b 2 4. (a-b)(a+b) = b(a-b) 5. a+b = b 6. 2b = b 7. 2=1 4 Slide 6 Proving Implication Many claims are in the form of If P, then Q, or, equivalently, P implies Q. For example: If ax 2 +bx+c = 0 and a 0, then x = (-b (b 2 - 4ac) 0.5 )/2a (Goldbachs Conjecture) If n is an even integer greater than 2, then n is a sum of two primes. 5 Slide 7 Proving Implication Method #1: Assume P. Show that Q logically follows. For example, prove that: If 0 x 2, then x 3 +4x+1 > 0 6 -x 3 +4x+1 = x(x 2 -4)+1 = x(x+2)(x-2)+1 Slide 8 Prove Implication Method #2 (Prove the Contrapositive) Assume not Q Show that not P logically follows For example: If r is irrational, then r 0.5 is also irrational. 7 r 0.5 = p/q, then r = p 2 /q 2 Slide 9 Proving If and Only If P if and only if Q For example: An integer is a multiple of 3 if and only if the sum of its digits is a multiple of 3. A positive integer p 2 is a prime if and only if 1+(p-1) (p-2)3 2 1 is a multiple of p. 8 Slide 10 Proving If and Only If Method #1: First, prove P implies Q.. Then, prove Q implies P. For example: A (B C) = (A B) (A C) 9 Slide 11 Proving If and Only If Method 2: Construct a chain of if-and-only-if implications. Show that P is equivalent to the second statement, which is equivalent to the third statement, , until reaching Q. For example: The standard deviation of a sequence of values is zero if and only if all the values are equal to the mean. 10 Slide 12 Existential Conditions Propositions with existential quantifiers: There is an object with a certain property such that something is true. We can use constructive method to prove these statements. The basic idea is to construct the object that makes the statement true. We only need one object to prove the statement true. 11 Slide 13 Existential Conditions For example: There exists an integer n that can be written in two ways as a sum of two primes. Can you construct one such integer? For example: There exists a unique prime number of the form n 2 1 where n is an integer 2. Can you construct one such prime? How to show the uniqueness? 12 (n-1)(n+2) Slide 14 Universal Conditions Propositions with universal quantifiers: For all x in the set S, a certain property on x is true. We can proceed in different ways: Construct a counter-example to show that the statement is false (doesnt work for all x in S). Start with an arbitrary element x in S and show that x satisfies the property. This is to show that the statement is true. 13 Slide 15 Class Discussion For example: For any positive real numbers x and y, floor(xy) = floor(x) floor(y) Which direction will you want to prove, true or false? How? 14 Consider x = 3.2, y = 2.9 Then floor(x y) = 0 But floor(x) floor(y) = 1 Slide 16 Proof by Contradiction The method of proof by contradiction is based on the fact that a statement can either be true or false, but not both. Suppose you can show that the assumption that a given statement P is not true will lead logically to a contradiction or an impossibility, we can conclude that the assumption is wrong, i.e., P is true. 15 Slide 17 Proof by Contradiction To do proof by contradiction, the first step is to assume that the statement to be proved is false. Then, show that this assumption leads logically to a contradiction. Then, we can conclude that the statement to be proved is true. 16 Slide 18 Proof by Contradiction For example: If n is an integer and n 2 is even, then n is even. First assume that an arbitrary integer n is given such that n 2 is even, but n is odd. Then, 17 Slide 19 Game with Contradiction Consider the following 3 statements: (1) There are 3 statements here. (2) Two of them are not true. (3) The average increase in IQ scores after taking CSCI3190 is more than 20 points. Is statement 3 true? 18 Slide 20 Summary of Proof Forms Direct proof of implication Contra-positive proof of implication Bi-directional proof of if and only if condition Chain of equivalence for if and only if condition Proof of existential statements Proof of universal statements Proof by contradiction 19